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- Title
Stretched exponential decay for subcritical parking times on Z<sup>d</sup>.
- Authors
Damron, Michael; Lyu, Hanbaek; Sivakoff, David
- Abstract
At each vertex of Z𝑑, place a car with probability p or vacant parking spot with probability 1-p. Cars perform independent random walks and park at vacant spots, rendering them passable. There is a transition at p=1/2: the origin is a.s. visited by finitely many distinct cars when p<1/2, and by infinitely many when p≥1/2. Furthermore, a.s. all cars park if p≤1/2 and some never park for p>1/2. For small p, we prove that the parking time 𝜏 of the car initially at the origin satisfies exp (-C1t 𝑑 𝑑+2) ≤ Pp(𝜏 > t) ≤ exp (-c2t 𝑑 𝑑+2). For d =1, these inequalities hold for p<1/2. In contrast, when p>1/2 and d =1, methods of Bramson–Lebowitz imply that the tail of the parking time of the spot of the origin decays like e-c √ t. Our exponent d/(d +2) also differs from those previously obtained in the case of moving obstacles.
- Subjects
RANDOM walks; AUTOMOBILES; PARKS; DOCKS
- Publication
Random Structures & Algorithms, 2021, Vol 59, Issue 2, p143
- ISSN
1042-9832
- Publication type
Article
- DOI
10.1002/rsa.21001